Here is what you do -- pressure folks at the end of a stun, not at the start of one.
What I mean is that after you get a stun with most characters you are obligated to either wait for a hold and throw it, or just mixup into a strike and yolo the situation. With Bayman its not quite the same. For him, you get a stun and then you are basically waiting for the end of the stun to happen. As the end of the stun happens, THAT is when you tap them with an offensive hold like the DDT or a strike like 236K (+1 on block) or 6P+K (+9), etc.
In this way, you control the situation and don't take as much risk. If they hold early, you smash them with a back breaker or an arm break. If they start mashing attacks out at the end of the stun, you'll be rewarded with a high-counter hold when your OH connects. If, for some reason, they are mashing a crush and it goes under the DDT then it will at the very worst be a normal hit attack as counter-hits against an offensive hold are impossible. It is also worth noting that unlike other characters, any time Bayman has one of his throws or holds broken he is at +3. So even if someone breaks your DDT, you keep right on fucking with them.
With Bayman, you can pace your in-stun offense easier than with other characters, and with far fewer repercussions when it backfires.
And if you DO decide to play for strikes inside the stun game for whatever reason, you are actually a much bigger bully than you might realize. 2kk is a 2-in-1 inside of stun and will fill out the threshold pretty much instantly. 1PP is another 2-in-1 inside of stun, but is instead good for knocking people into the environment when you have them cornered (the first hit tracks).
From range, 8PP is a 2-in-1 launch as well, but only on normal hit. This means you use this string strictly as whiff punishment. It has deceptively killer range, so make sure to abuse it.
He is also one of the few characters with a good CH high jab that lets him truly start up. PK both will connect on CH, which gives him a long enough stun that it lets him mix up into his followup PKP for backturned pressure shenanigans, or PKK for a guaranteed ground throw.